A robust ultrasensitive transcriptional switch in noisy cellular environments

Ultrasensitive transcriptional switches enable sharp transitions between transcriptional on and off states and are essential for cells to respond to environmental cues with high fidelity. However, conventional switches, which rely on direct repressor-DNA binding, are extremely noise-sensitive, leading to unintended changes in gene expression. Here, through model simulations and analysis, we discovered that an alternative design combining three indirect transcriptional repression mechanisms, sequestration, blocking, and displacement, can generate a noise-resilient ultrasensitive switch. Although sequestration alone can generate an ultrasensitive switch, it remains sensitive to noise because the unintended transcriptional state induced by noise persists for long periods. However, by jointly utilizing blocking and displacement, these noise-induced transitions can be rapidly restored to the original transcriptional state. Because this transcriptional switch is effective in noisy cellular contexts, it goes beyond previous synthetic transcriptional switches, making it particularly valuable for robust synthetic system design. Our findings also provide insights into the evolution of robust ultrasensitive switches in cells. Specifically, the concurrent use of seemingly redundant indirect repression mechanisms in diverse biological systems appears to be a strategy to achieve noise-resilience of ultrasensitive switches.


of the three binding sites model with cooperative binding
We used an approach to calculate the transcriptional activity and the Fano factor is established by A. Sanchez et al 1 .Specifically, the transcription regulated by cooperative binding with three binding sites on DNA (Fig. 1a and Supplementary Table 1) can be described by the following CMEs: where 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 ⎦ Here,  ! is a probability vector whose components  !& represent the joint probability that DNA is in state  & with  mRNAs,  ∈ {000, 001, 010, 100, 011, 101, 110, 111} .The stochastic transitions between DNA states are described by the matrix , where  ' is the total number of repressors,  ( and  ) are the association and dissociation rates between repressors and DNA, respectively, and  is cooperativity.Additionally,  is the diagonal matrix whose diagonal elements represent the production rate of mRNA in each DNA state.
Specifically, the last diagonal element of  is zero, because the transcription is suppressed only when all three binding sites are occupied by repressors (i.e., the state  ### ).On the other hand, the remaining diagonal elements of  are , which is the production rate of mRNA, as the transcription is activated in the remaining DNA states.Each mRNA is degraded with the rate .Hence,  !decreases with the rate  ! and increases with the rate ( + 1) !"# in Eq. (1).
To reduce the number of parameters, we rescaled time variable  by  (  ' in Eq. (1) as follows: where Here,  ) =  ) / ( is the dissociation constant between repressors and free DNA, is the effective total number of the repressor,  C = / (  ' and  B = / (  ' are normalized production and degradation rates of mRNA, respectively. Then, from Eq. ( 2), we can calculate the th moment vector of mRNAs at the steady state, , whose elements represent the th moment of mRNAs in each DNA state.

Supplementary Note 2. The equations for the transcriptional activity and the Fano factor of the model describing the sequestration
The transcription regulated by the sole sequestration (Fig. 2a and Supplementary Table 1) can be described using Eq. ( 1) with where  !& is the joint probability that DNA is in state  & with  mRNAs,  ∈ {, },  ( and  : are the association and dissociation rates between free activator  and DNA, respectively. To calculate the stationary mean and variance of mRNAs in this system, we first derived the stationary mean of , 〈〉, in the matrix .Specifically, 〈〉 can be derived from the CME describing the reversible binding (i.e., sequestration) between  and free repressor , where  = is the dissociation constant between  and .
As the total numbers of the activator ( ' =  +  < +  < ) and the repressor ( ' =  +  < ) are conserved and the total number of the activator is much larger than that of DNA (i.e.,  ' ≫ 1),  < and  in Eq. ( 10) can be replaced with  ' −  −  < ≈  ' −  and  ' − ( ' −  −  < ) ≈  ' − ( ' − ).Consequently, we can get the equation for the first and the second moments of  as follows: Assuming that 〈 / 〉 ≈ 〈〉 / (i.e., () = 〈 / 〉 − 〈〉 / ≈ 0), Eq. ( 11) can be switched to the quadratic equation for 〈〉.Then, we can get the approximated equation for 〈〉, and thus the approximated equations for 〈〉 and 〈 < 〉 as follows: These approximations are accurate as long as the number of the activator is sufficiently large Here,  : =  : / ( is the dissociation constant between free activator  and DNA,  B = / ' ,  ∈ {,  ' ,  = ,  : },  C = / (  ' , and  B = / (  ' .Then, by solving the steady state equation 0 =  A  % and Eq. ( 5), we can get The transcriptional activity, N ' E O, is defined as the probability that DNA is activated, and thus it is the second element of  % (Supplementary Table 2): Subsequently, substituting  % and  # to Eq. (8) and Eq. ( 9), we can get the following approximations for 〈〉 and 〈 / 〉 as follows: and Using these, we can obtain the Fano factor of  , N ' E O , defined as (Supplementary Table 2) as follows:

Supplementary Note 3. The equations for the transcriptional activity and the Fano factor of the model describing the sequestration and blocking
The transcription regulated by the combination of the sequestration and blocking (Fig. 3a and Supplementary Table 1) can be described using Eq. ( 1) with where  !& is the joint probability that DNA is in state  & with  mRNAs,  ∈ {, , },  ( and  M are the association and dissociation rates between free repressor  and the DNAbound activator, respectively.As done in the model describing the sole sequestration, we derived the stationary means of , , and  < as in Eq. (12).Then, rescaling time variable  by  (  ' , we can rewrite the CME describing the sequestration and blocking as Eq. ( 2), where The transcriptional activity, N ' E O, is defined as the probability that DNA is activated, and thus it is the second element of  % (Supplementary Table 2): Similarly, by solving Eq. ( 5), we can get Subsequently, substituting  % and  # to Eq. (8) and Eq.(9), we can get the following approximations for 〈〉 and 〈 / 〉 as follows: and Using these, we can obtain the Fano factor of  , N ' E O , defined as (Supplementary Table 2) as follows: .

Supplementary Note 4. The equations for the transcriptional activity and the Fano factor of the model describing the sequestration, blocking, and displacement
The transcription regulated by the combination of the sequestration, blocking, and displacement (Fig. 4a and Supplementary Table 1) can be described using Eq. ( 1) with where  !& is the joint probability that DNA is in state  & with  mRNAs,  ∈ {, , },  ( and  R are the association and dissociation rates between the repressor-activator complex  < and DNA, respectively.As done in models describing the sole sequestration and describing the sequestration and blocking, we derived the stationary means of , , and  < as in Eq. (12).Then, rescaling time variable  by  (  ' , we can rewrite the CME describing the sequestration, blocking, and displacement as Eq. ( 2), where Here,  R =  M / ( is the dissociation constant between the repressor-activator complex  < and DNA,  B = / ' ,  ∈ {, ,  < ,  ' ,  = ,  : ,  M ,  R },  C = / (  ' , and  B = / (  ' .Then, by solving the steady state equation 0 =  A  % , we can get The transcriptional activity, N ' E O, is defined as the probability that DNA is activated, and thus it is the second element of  % (Supplementary Table 2): Similarly, by solving Eq. ( 5), we can get where Subsequently, substituting  % and  # to Eq. (8) and Eq. ( 9), we can get the following approximations for 〈〉 and 〈 / 〉 as follows: and Using these, we can obtain the Fano factor of  , N ' E O , defined as (Supplementary Table 2) as follows: .
Supplementary Table 1.Propensity functions of reactions and parameter values for all models.
The sequestration-based switch Transcriptional Fano factor The sequestration-and blocking-based switch sequestration model Transcriptional , where  ( 0 Fano factor where Fano factor   -4, the number of the total repressors ( ' ) is fixed.b These switches can exhibit the same mean of mRNAs with respect to  ' by modifying the value of  M E (Supplementary Table 1), resulting in the same sensitivity.c On the other hand, the switch utilizing sequestration, blocking, and displacement shows the lowest variance in the number of mRNAs among the three types of switches.d, e Consequently, both the Fano factor (d) and the coefficient of variation (CV), defined as the standard deviation over the mean (e), are also lower for the triple-mechanism switch than for the others.f When birth-death reactions of R q with rates  D w and  D E are incorporated into the ultrasensitive switches,  ' fluctuates rather than remains fixed.represents the probability density function of the Poisson distribution with the mean of  D w/ D E .Then, the calculated mean of mRNAs is identical in the three types of switches (g).On the other hand, the switch utilizing sequestration, blocking, and displacement shows the least variance of mRNAs among the three types of switches (h).i, j As a result, the Fano factor (i) and CV (j) are also lower for the triple-mechanism switch than for the others.Here,  D E = 10 $g and  ' = 100 are used, and  D w is adjusted to vary the value of 〈 ' E 〉.
Supplementary Figure 3.The transcriptional switch utilizing all three mechanismssequestration, blocking, and displacement -can generate ultrasenstivity robust to noise.a In the sole sequestration-based switch, strong sequestration and strong activators (i.e.,  = E <  : E and  : E < 10 $# ) are required to generate ultrasensitive transcriptional activity.The sensitivity of the transcriptional activity is quantified using the effective Hill coefficient.The black line denotes the effective Hill coefficient of 10. b When the effective Hill coefficient exceeds 10 (i.e., when ultrasensitivity is generated; below the black line), transcriptional noise is low during TRP (top), but high during TAP (bottom).Transcriptional noise is quantified by the normalized area of the plot for the Fano factor with respect to  ' E (e.g., Fig. 2b(ii)) on [10 $# , 10 $%.# ] and [10 %.# , 10 # ] during TRP and TAP, respectively).c With the addition of weak blocking ( M E = 10 $# ) to the sole sequestration-based switch, the sensitivity remains comparable to that generated by the sole sequestration-based switch.d Concurrently, the transcriptional noise during TRP is reduced (bottom).e-f Stronger blocking ( M E = 10 $. ) further diminishes transcriptional noise during TRP (f, bottom), but elevates it during TAP (f, top).g-h When the blocking becomes too strong ( M E = 10 $y ), ultrasensitivity is not generated.i After the addition of weak displacement ( R E = 10 $. ) to the sequestrationand blocking-based switch with  M E = 10 $. , ultrasensitivity is generated within a narrower range.j In this range, the transcriptional noise during TAP is slightly higher (top), while during TRP it remains comparable (bottom) to that generated by the sequestration-and blocking-based switch (f).k-l Stronger displacement ( R E = 10 $# ) can generate ultrasensitivity over a broader range (k), while reducing the transcriptional noise during TAP (l, top) compared to the sequestration-and blocking-based switch (f).m-n Moreover, as further stronger displacement ( R E = 10 # ) is added (m), ultrasensitivity is generated over a broader range and the transcriptional noise is more reduced during TAP (n, top).

Supplementary Figure 1 .Supplementary Figure 2 .
The stationary distributions of mRNAs transcribed during TRP and TAP in each transcriptional ultrasensitive switch.a In the sequestration-based switch, the stationary distribution of mRNAs shows an additional peak far from zero, leading to bimodality with a strong activator (blue bars) unlike with a weak activator (red bars) during TRP.b In contrast, in the sequestration-and blocking-based switch, the stationary distribution of mRNAs is concentrated around zero (blue bars) without bimodality observed in sole sequestration (red bars).c However, during TAP, it shows an additional peak around zero, leading to bimodality with blocking (blue bars) unlike with sequestration alone (red bars).d In the sequestration-, blocking-, and displacement-based switch, such an additional peak around zero is removed from the stationary distribution of mRNAs during TAP.The sequestration, blocking, and displacement-based switch exhibits superior robustness to noise compared to the other two types of switches, even when the means of the mRNAs are identical and the total number of repressors fluctuates.a In the three types of transcriptional ultrasensitive switches used in Figs. 2